Derivative of \( \displaystyle - \ln{\left(\cos{\left(x - 1 \right)} \right)} \)
Problem 2.1769 · hard
Differentiate \( \displaystyle f(x) = - \ln{\left(\cos{\left(x - 1 \right)} \right)} \).
- \[ \frac{d}{d x} \left(- \ln{\left(\cos{\left(x - 1 \right)} \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} \ln{\left(\cos{\left(x - 1 \right)} \right)} \]constantPull out the negative sign.✓ Proved
- \[ = - \frac{\frac{d}{d x} \cos{\left(x - 1 \right)}}{\cos{\left(x - 1 \right)}} \]logarithmicApply the derivative rule for the natural logarithm.✓ Proved
- \[ = \frac{\sin{\left(x - 1 \right)}}{\cos{\left(x - 1 \right)}} \]trig algebraDifferentiate the cosine function. Simplify the signs and the fraction.✓ Proved
- \[ = \tan{\left(x - 1 \right)} \]simplifyUse the identity tan(u) = sin(u)/cos(u).✓ Proved
Answer \( \tan{\left(x - 1 \right)} \)
Mind the domain. The answer is also defined at points where f(x) is not. Substituting there gives a number that is not a slope of f.
Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where cos(x - 1) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(x - 1) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(x - 1) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(x - 1) = 0 tan has poles at odd multiples of pi/2 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — Step 2 uses the label 'constant' to pull out the negative sign, but the rule for factoring out a constant multiplier is 'constant-multiple'. Step 4 uses the label 'trig' to perform the differentiation of cos(x-1), but 'trig' is not a differentiation rule; this step requires 'derivative' (for the cosine form) and 'chain' (for the inner function), or at minimum 'derivative' if the chain rule is considered part of the derivative lookup, but 'trig' is not in the allowed vocabulary for differentiation steps.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — Step 2 uses the label 'constant' to pull out the negative sign, but the rule for factoring out a constant multiplier is 'constant-multiple'. Step 4 uses the label 'trig' to perform the differentiation of cos(x-1), but 'trig' is not a differentiation rule; this step requires 'derivative' (for the cosine form) and 'chain' (for the inner function), or at minimum 'derivative' if the chain rule is considered part of the derivative lookup, but 'trig' is not in the allowed vocabulary for differentiation steps.gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — Step 4 applies the chain rule to differentiate cos(x-1) but labels it 'trig'. The 'trig' label is reserved for applying trigonometric identities or derivatives of basic trig functions without an inner function chain, whereas this step requires the chain rule (or 'chain' label) to handle the inner derivative of (x-1). Additionally, Step 3 applies the chain rule for the logarithm but labels it 'logarithmic', which is acceptable, but Step 4's omission of the chain rule label for the inner derivative is a defect in naming the rule applied.
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
gemma4:26b, checked 2026-10-07 with SymPy 1.14.0.